Properties

Label 2070.2.a.x
Level $2070$
Weight $2$
Character orbit 2070.a
Self dual yes
Analytic conductor $16.529$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} + \beta q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + q^{5} + \beta q^{7} + q^{8} + q^{10} + ( -1 - \beta ) q^{11} + ( 4 - \beta ) q^{13} + \beta q^{14} + q^{16} + ( 1 + \beta ) q^{17} + ( 4 - \beta ) q^{19} + q^{20} + ( -1 - \beta ) q^{22} - q^{23} + q^{25} + ( 4 - \beta ) q^{26} + \beta q^{28} + ( -4 + 2 \beta ) q^{29} + ( 2 + 3 \beta ) q^{31} + q^{32} + ( 1 + \beta ) q^{34} + \beta q^{35} -4 q^{37} + ( 4 - \beta ) q^{38} + q^{40} + ( 5 - \beta ) q^{41} + ( 4 - 4 \beta ) q^{43} + ( -1 - \beta ) q^{44} - q^{46} + ( 8 + 2 \beta ) q^{47} + ( -2 + \beta ) q^{49} + q^{50} + ( 4 - \beta ) q^{52} -6 q^{53} + ( -1 - \beta ) q^{55} + \beta q^{56} + ( -4 + 2 \beta ) q^{58} + ( 10 - 2 \beta ) q^{59} + ( 5 - 3 \beta ) q^{61} + ( 2 + 3 \beta ) q^{62} + q^{64} + ( 4 - \beta ) q^{65} + ( 4 - 4 \beta ) q^{67} + ( 1 + \beta ) q^{68} + \beta q^{70} + ( -3 + 3 \beta ) q^{71} + ( 2 - 6 \beta ) q^{73} -4 q^{74} + ( 4 - \beta ) q^{76} + ( -5 - 2 \beta ) q^{77} + 8 q^{79} + q^{80} + ( 5 - \beta ) q^{82} + 6 q^{83} + ( 1 + \beta ) q^{85} + ( 4 - 4 \beta ) q^{86} + ( -1 - \beta ) q^{88} + ( -8 + 4 \beta ) q^{89} + ( -5 + 3 \beta ) q^{91} - q^{92} + ( 8 + 2 \beta ) q^{94} + ( 4 - \beta ) q^{95} + ( 1 + 5 \beta ) q^{97} + ( -2 + \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + q^{7} + 2q^{8} + 2q^{10} - 3q^{11} + 7q^{13} + q^{14} + 2q^{16} + 3q^{17} + 7q^{19} + 2q^{20} - 3q^{22} - 2q^{23} + 2q^{25} + 7q^{26} + q^{28} - 6q^{29} + 7q^{31} + 2q^{32} + 3q^{34} + q^{35} - 8q^{37} + 7q^{38} + 2q^{40} + 9q^{41} + 4q^{43} - 3q^{44} - 2q^{46} + 18q^{47} - 3q^{49} + 2q^{50} + 7q^{52} - 12q^{53} - 3q^{55} + q^{56} - 6q^{58} + 18q^{59} + 7q^{61} + 7q^{62} + 2q^{64} + 7q^{65} + 4q^{67} + 3q^{68} + q^{70} - 3q^{71} - 2q^{73} - 8q^{74} + 7q^{76} - 12q^{77} + 16q^{79} + 2q^{80} + 9q^{82} + 12q^{83} + 3q^{85} + 4q^{86} - 3q^{88} - 12q^{89} - 7q^{91} - 2q^{92} + 18q^{94} + 7q^{95} + 7q^{97} - 3q^{98} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.79129
2.79129
1.00000 0 1.00000 1.00000 0 −1.79129 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 2.79129 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2070.2.a.x 2
3.b odd 2 1 230.2.a.a 2
12.b even 2 1 1840.2.a.n 2
15.d odd 2 1 1150.2.a.o 2
15.e even 4 2 1150.2.b.g 4
24.f even 2 1 7360.2.a.bk 2
24.h odd 2 1 7360.2.a.bq 2
60.h even 2 1 9200.2.a.bs 2
69.c even 2 1 5290.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.a 2 3.b odd 2 1
1150.2.a.o 2 15.d odd 2 1
1150.2.b.g 4 15.e even 4 2
1840.2.a.n 2 12.b even 2 1
2070.2.a.x 2 1.a even 1 1 trivial
5290.2.a.e 2 69.c even 2 1
7360.2.a.bk 2 24.f even 2 1
7360.2.a.bq 2 24.h odd 2 1
9200.2.a.bs 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2070))\):

\( T_{7}^{2} - T_{7} - 5 \)
\( T_{11}^{2} + 3 T_{11} - 3 \)
\( T_{13}^{2} - 7 T_{13} + 7 \)
\( T_{17}^{2} - 3 T_{17} - 3 \)
\( T_{29}^{2} + 6 T_{29} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -5 - T + T^{2} \)
$11$ \( -3 + 3 T + T^{2} \)
$13$ \( 7 - 7 T + T^{2} \)
$17$ \( -3 - 3 T + T^{2} \)
$19$ \( 7 - 7 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -12 + 6 T + T^{2} \)
$31$ \( -35 - 7 T + T^{2} \)
$37$ \( ( 4 + T )^{2} \)
$41$ \( 15 - 9 T + T^{2} \)
$43$ \( -80 - 4 T + T^{2} \)
$47$ \( 60 - 18 T + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 60 - 18 T + T^{2} \)
$61$ \( -35 - 7 T + T^{2} \)
$67$ \( -80 - 4 T + T^{2} \)
$71$ \( -45 + 3 T + T^{2} \)
$73$ \( -188 + 2 T + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( -48 + 12 T + T^{2} \)
$97$ \( -119 - 7 T + T^{2} \)
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